\section{Laplacian's Adjoint Operator}

Just a brief derivation of Laplaician operator's adjoint operator with respect to $\textit{L}^2$ inner product space \textbf{U} and \textbf{V} and certain boundary condtion to be mentioned later. Such inner product is defined just as

\begin{equation}\nonumber
L^2<u,v> = \int_{\Omega} uv d\Omega = \int_{\Omega}<u, v>d\Omega, \forall u \in \textbf{U}, \forall v \in \textbf{V}
\end{equation}

Basically we have the following identity

\begin{equation}\nonumber
L^2<\nabla^2u, v> = L^2<u, \nabla^{\star}v>
\end{equation}

and we want to figure out what $\nabla^{\star}$ is. We simply start from the left handside

\begin{equation}\nonumber
\begin{split}
L^2<\nabla^2u, v> &= \int_{\Omega}<\nabla^2u, v> d\Omega = \int_{\Omega}<\nabla\cdot\nabla u, v> d\Omega\\
&= \oint_{\partial\Omega}v<\nabla u, \vec{\textbf{n}}> ds - \int_{\Omega}<\nabla u, \nabla v>d\Omega\\
&= \oint_{\partial\Omega}v<\nabla u, \vec{\textbf{n}}> ds - (\oint_{\partial\Omega}u<\nabla v, \vec{\textbf{n}}> ds - \int_{\Omega}<u, \nabla^2 v>d\Omega)\\
&= \oint_{\partial\Omega}(v<\nabla u, \vec{\textbf{n}}> - u<\nabla v, \vec{\textbf{n}}>) ds + \int_{\Omega}<u, \nabla^2 v>d\Omega
\end{split}
\end{equation}

If the boundary integral vanishes, which means

\begin{equation}\nonumber
\oint_{\partial\Omega}(v<\nabla u, \vec{\textbf{n}}> - u<\nabla v, \vec{\textbf{n}}>) ds = 0
\end{equation}

we have

\begin{equation}\nonumber
L^2<\nabla^2u, v> = \int_{\Omega}<u, \nabla^2 v>d\Omega = L^2<u, \nabla^2 v>
\end{equation}

which means $\nabla^{\star} = \nabla^2$. The multivariable version of integration by part we use above is actually extension of divergence theorem

\begin{equation}\nonumber
\begin{split}
\int_{\Omega}\nabla \cdot \vec{\textbf{u}} d\Omega = \oint_{\partial\Omega} <\vec{\textbf{u}}, \vec{\textbf{n}}> ds, \nabla \cdot (v\vec{\textbf{u}}) = <\vec{\textbf{u}}, \nabla v> + <\nabla\cdot\vec{\textbf{u}}, v>\\
\Rightarrow \int_{\Omega} \nabla \cdot (v\vec{\textbf{u}}) d\Omega = \int_{\Omega} (<\vec{\textbf{u}}, \nabla v> + <\nabla\cdot\vec{\textbf{u}}, v>)d\Omega =\oint_{\partial\Omega}v<\vec{\textbf{u}},\vec{\textbf{n}}>ds\\
\Rightarrow \int_{\Omega} <\vec{\textbf{u}}, \nabla v>d\Omega + \int_{\Omega}<\nabla\cdot\vec{\textbf{u}}, v>d\Omega = \oint_{\partial\Omega}v<\vec{\textbf{u}},\vec{\textbf{n}}>ds
\end{split}
\end{equation}

which also tells us that $-\nabla$ is the adjoint operator of $\nabla\cdot$ with respect to $\textit{L}^2$ inner product space with appropriate boundary condition.